Applications of Integrals in Economics

The concept of integration is widely used in business and economics. In this section, we consider the following applications of integrals in finance and economics:

• Marginal and total revenue, cost, and profit;
• Capital accumulation over a specified period of time;
• Consumer and producer surplus;
• Lorenz curve and Gini coefficient;

Marginal and Total Revenue, Cost, and Profit

Marginal revenue $$\left({MR}\right)$$ is the additional revenue gained by producing one more unit of a product or service.

It can also be described as the change in total revenue $$\left({TR}\right)$$ divided by the change in number of units sold $$\left({Q}\right):$$

$MR = \frac{{dTR}}{{dQ}}.$

If a marginal revenue function $$MR\left( Q \right)$$ is known, the total revenue can be obtained by integrating the marginal revenue function:

$TR\left( Q \right) = \int {MR\left( Q \right)dQ} ,$

where integration is carried out over a certain interval of $$Q.$$

Marginal cost $$\left({MC}\right)$$ denotes the additional cost of producing one extra unit of output.

The similar relationship exists between the marginal cost $$MC$$ and the total cost $$TC:$$

$MC = \frac{{dTC}}{{dQ}},$

so

$TC\left( Q \right) = \int {MC\left( Q \right)dQ} .$

Since profit is defined as

$TP = TR - TC,$

we can write the following equation for marginal profit $$\left({MP}\right):$$

$MP = MR - MC,\;\;\text{or}\;\; \frac{{dTP}}{{dQ}} = \frac{{dTR}}{{dQ}} - \frac{{dTC}}{{dQ}}.$

Capital Accumulation Over a Period

Let $$I\left( t \right)$$ be the rate of investment. The total capital accumulation $$K$$ during the time interval $$\left[ {a,b} \right]$$ can be estimated by the formula

$K = \int\limits_a^b {I\left( t \right)dt} .$

Consumer and Producer Surplus

The demand function or demand curve shows the relationship between the price of a certain product or service and the quantity demanded over a period of time.

The supply function or supply curve shows the quantity of a product or service that producers will supply over a period of time at any given price.

Both these price-quantity relationships are usually considered as functions of quantity $$\left( Q \right).$$

Generally, the demand function $$P = D\left( Q \right)$$ is decreasing, because consumers are likely to buy more of a product at lower prices. Unlike the law of demand, the supply function $$P = S\left( Q \right)$$ is increasing, because producers are willing to deliver a greater quantity of a product at higher prices.

The point $$\left( {{Q_0},{P_0}} \right)$$ where the demand and supply curves intersect is called the market equilibrium point.

The maximum price a consumer is willing and able to pay is defined by the demand curve $$P = D\left( Q \right).$$ For quantities $${Q \lt {Q_0}},$$ it is greater than the equilibrium price $${P_0}$$ in the market. Consumers gain by buying at the equilibrium price rather than at a higher price. This net gain is called consumer surplus.

Consumer surplus is represented by the area under the demand curve $$P = D\left( Q \right)$$ and above the horizontal line $$P = {P_0}$$ at the level of the market price.

Consumer surplus $$\left( {CS} \right)$$ is thus defined by the integration formula

$CS = \int\limits_0^{{Q_0}} {D\left( Q \right)dQ} - {P_0}{Q_0} = \int\limits_0^{{Q_0}} {\left[ {D\left( Q \right) - {P_0}} \right]dQ} .$

A similar analysis shows that producers also gain if they trade their products at the market equilibrium price. Their gain is called producer surplus $$\left( {PS} \right)$$ and is given by the equation

$PS = {P_0}{Q_0} - \int\limits_0^{{Q_0}} {S\left( Q \right)dQ} = \int\limits_0^{{Q_0}} {\left[ {{P_0} - S\left( Q \right)} \right]dQ} .$

Lorenz Curve and Gini Coefficient

The Lorenz curve is a graphical representation of income or wealth distribution among a population.

The horizontal axis on a Lorenz curve typically shows the portion or percentage of total population, and the vertical axis shows the portion of total income or wealth. For instance, if a Lorenz curve has a point with coordinates $$\left( {0.4,0.2} \right),$$ this means that the first $$40\%$$ of population (ranked by income in increasing order) earned $$20\%$$ of total income.

The Lorenz Curve is represented by a convex curve. A more convex Lorenz curve implies more inequality in income distribution. The area between the $$45-$$degree line (the line of equality) and the Lorenz curve can be used as a measure of inequality.

The Gini coefficient $$G$$ is defined as the area between the line of equality and the Lorenz curve, divided by the total area under the line of equality:

$G = \frac{A}{{A + B}} = 2\int\limits_0^1 {\left[ {x - L\left( x \right)} \right]dx} .$

The Gini coefficient is a relative measure of inequality. It ranges from $$0$$ (or $$0\%$$) to $$1$$ (or $$100\%$$), with $$0$$ representing perfect equality in a population and $$1$$ representing perfect inequality.

Solved Problems

Click or tap a problem to see the solution.

Example 1

The marginal revenue of a company is given by $MR = 100 + 20Q + 3{Q^2},$ where $$Q$$ is amount of units sold for a period. Find the total revenue function if at $$Q = 2$$ it is equal to $$260.$$

Example 2

The rate of investment is given by $I\left( t \right) = 6\sqrt t .$ Calculate the capital growth between the $$4^{\text{th}}$$ and the $$9^{\text{th}}$$ years.

Example 3

Assume the rate of investment is given by the function $I\left( t \right) = \ln t .$ Compute the total capital accumulation between the $$1^{\text{st}}$$ and the $$5^{\text{th}}$$ years.

Example 4

For a certain product, the demand function is $D\left( Q \right) = 1000 - 25Q,$ and the supply function is $S\left( Q \right) = 100 + {Q^2}.$ Compute the consumer and producer surplus.

Example 1.

The marginal revenue of a company is given by $MR = 100 + 20Q + 3{Q^2},$ where $$Q$$ is amount of units sold for a period. Find the total revenue function if at $$Q = 2$$ it is equal to $$260.$$

Solution.

We find the total revenue function $$TR$$ by integrating the marginal revenue function $$MR:$$

$TR\left( Q \right) = \int {MR\left( Q \right)dQ} = \int {\left( {100 + 20Q + 3{Q^2}} \right)dQ} = 100Q + 10{Q^2} + {Q^3} + C.$

The constant of integration $$C$$ can be determined using the initial condition $$TR\left( {Q = 2} \right) = 260.$$ Hence,

$200 + 40 + 8 + C = 260,\;\; \Rightarrow C = 12.$

So, the total revenue function is given by

$TR\left( Q \right) = 100Q + 10{Q^2} + {Q^3} + 12.$

Example 2.

The rate of investment is given by $I\left( t \right) = 6\sqrt t .$ Calculate the capital growth between the $$4^{\text{th}}$$ and the $$9^{\text{th}}$$ years.

Solution.

Using the integration formula

$K = \int\limits_a^b {I\left( t \right)dt} ,$

we have

$K = \int\limits_4^9 {6\sqrt t dt} = 6\int\limits_4^9 {{t^{\frac{1}{2}}}dt} = \left. {\frac{{12{t^{\frac{3}{2}}}}}{3}} \right|_4^9 = \left. {4{{\left( {\sqrt t } \right)}^3}} \right|_4^9 = 4\left( {{3^3} - {2^3}} \right) = 76.$

Example 3.

Assume the rate of investment is given by the function $I\left( t \right) = \ln t .$ Compute the total capital accumulation between the $$1^{\text{st}}$$ and the $$5^{\text{th}}$$ years.

Solution.

To calculate the capital accumulation, we use the formula

$K = \int\limits_a^b {I\left( t \right)dt} = \int\limits_1^5 {\ln tdt} .$

Integrating by parts, we have

$\int {\ln tdt} = \left[ {\begin{array}{*{20}{l}} {u = \ln t}\\ {dv = dt}\\ {du = \frac{{dt}}{t}}\\ {v = t} \end{array}} \right] = t\ln t - \int {\cancel{t}\frac{{dt}}{\cancel{t}}} = t\ln t - \int {dt} = t\ln t - t.$

Hence

$K = \left. {\left( {t\ln t - t} \right)} \right|_1^5 = \left( {5\ln 5 - 5} \right) - \left( {\ln 1 - 1} \right) = 5\ln 5 - 4 \approx 4.05$

Example 4.

For a certain product, the demand function is $D\left( Q \right) = 1000 - 25Q,$ and the supply function is $S\left( Q \right) = 100 + {Q^2}.$ Compute the consumer and producer surplus.

Solution.

First we determine the equilibrium point by equating the demand and supply functions:

$D\left( Q \right) = S\left( Q \right),\;\; \Rightarrow 1000 - 25Q = 100 + {Q^2},\;\; \Rightarrow {Q^2} + 25Q - 900 = 0.$

The positive solution of the quadratic equation is $${Q_0} = 20.$$ The market equilibrium price is $${P_0} = 500.$$

The consumer surplus $$CS$$ is given by

$CS = \int\limits_0^{{Q_0}} {\left[ {D\left( Q \right) - {P_0}} \right]dQ} = \int\limits_0^{20} {\left( {1000 - 25Q - 500} \right)dQ} = \int\limits_0^{20} {\left( {500 - 25Q} \right)dQ} = \left. {\left( {500Q - \frac{{25{Q^2}}}{2}} \right)} \right|_0^{20} = 10000 - 5000 = 5000.$

Similarly we find the producer surplus $$PS:$$

$PS = \int\limits_0^{{Q_0}} {\left[ {{P_0} - S\left( Q \right)} \right]dQ} = \int\limits_0^{20} {\left( {500 - 100 - {Q^2}} \right)dQ} = \int\limits_0^{20} {\left( {400 - {Q^2}} \right)dQ} = \left. {\left( {400Q - \frac{{{Q^3}}}{3}} \right)} \right|_0^{20} \approx 8000 - 2667 = 5333.$